Approximating Betweenness Centrality
David A. Bader, Shiva Kintali, Kamesh Madduri, and Milena Mihail
College of Computing
Georgia Institute of Technology
{bader,kintali,kamesh,mihail}@cc.gatech.edu
Abstract. Betweenness is a centrality measure based on shortest paths,
widely used in complex network analysis. It is computationally-expensive
to exactly determine betweenness; currently the fastest-known algorithm
by Brandes requires O(nm) time for unweighted graphs and O(nm +
n2 log n) time for weighted graphs, where n is the number of vertices
and m is the number of edges in the network. These are also the worstcase time bounds for computing the betweenness score of a single vertex.
In this paper, we present a novel approximation algorithm for computing betweenness centrality of a given vertex, for both weighted and unweighted graphs. Our approximation algorithm is based on an adaptive
sampling technique that significantly reduces the number of single-source
shortest path computations for vertices with high centrality. We conduct
an extensive experimental study on real-world graph instances, and observe that our random sampling algorithm gives very good betweenness
approximations for biological networks, road networks and web crawls.
1
Introduction
One of the fundamental problems in network analysis is to determine the importance (or the centrality) of a particular vertex (or an edge) in a network. Some
of the well-known metrics for computing centrality are closeness [1], stress [2]
and betweenness [3, 4]. Of these indices, betweenness has been extensively used
in recent years for the analysis of social-interaction networks, as well as other
large-scale complex networks. Some applications include lethality in biological
networks [5–7], study of sexual networks and AIDS [8], identifying key actors in
terrorist networks [9, 10], organizational behavior [11], and supply chain management processes [12]. Betweenness is also used as the primary routine in popular
algorithms for clustering and community identification [13] in real-world networks. For instance, the Girvan-Newman [14] algorithm iteratively partitions a
network by identifying edges with high betweenness scores, removing them and
recomputing centrality scores.
Betweenness is a global centrality metric that is based on shortest-path enumeration. Consider a graph G = (V, E), where V is the set of vertices representing actors or nodes in the complex network, and E, the set of edges representing
the relationships between the vertices. The number of vertices and edges are
denoted by n and m respectively. The graphs can be directed or undirected.
We will assume that each edge e ∈ E has a positive integer weight w(e). For
unweighted graphs, we use w(e) = 1. A path from vertex s to t is defined as a
sequence of edges hui , ui+1 i, 0 ≤ i < l, where u0 = s and ul = t. The length of
a path is the sum of the weights of edges. We use d(s, t) to denote the distance
between vertices s and t (the minimum length of any path connecting s and t
in G). Let us denote the total number of shortest paths between vertices s and
t by λst , and the number passing through vertex v by λst (v). Let δst (v) denote
the f raction of shortest paths between s and t that pass through a particular
(v)
vertex v i.e., δst (v) = λst
λst . We call δst (v) the pair-dependency of s, t on v.
Betweenness centrality [3, 4] of a vertex v is defined as
BC(v) =
X
δst (v)
s6=v6=t∈V
Currently, the fastest known algorithm for exactly computing betweenness
of all the vertices, designed by Brandes [15], requires at least O(nm) time for
unweighted graphs and O(nm + n2 log n) time for weighted graphs, where n
is the number of vertices and m is the number of edges. Thus, for large-scale
graphs, exact centrality computation on current workstations is not practically
viable. In prior work, we explored high performance computing techniques [16]
that exploit the typical small-world graph topology to speed up exact centrality
computation. We designed novel parallel algorithms to exactly compute various
centrality metrics, optimized for real-world networks. We also demonstrate the
capability to compute exact betweenness on several large-scale networks (vertices
and edges in the order of millions) from the Internet and social interaction data;
these networks are three orders of magnitude larger than instances that can be
processed by current social network analysis packages.
Fast centrality estimation is thus an important problem, as a good approximation would be an acceptable alternative to exact scores. Currently the fastest
exact algorithms for shortest path enumeration-based metrics require n shortestpath computations; however, it is possible to estimate centrality by extrapolating scores from a fewer number of path computations. Using a random sampling
technique, Eppstein and Wang [17] show that the closeness centrality of all vertices in a weighted, undirected graph can be approximated with high probability
n
in O( log
²2 (n log n + m)) time, and an additive error of at most ²∆G (² is a fixed
constant, and ∆G is the diameter of the graph). However, betweenness centrality scores are harder to estimate, and the quality of approximation is found to
be dependent on the vertices from which the shortest path computations are
initiated from (in this paper, we will refer to them as the set of source vertices
for the approximation algorithm). Recently, Brandes and Pich [18] presented
centrality estimation heuristics, where they experimented with different strategies for selecting the source vertices. They observe that a random selection of
source vertices is superior to deterministic strategies. In addition to exact parallel algorithms, we also discussed parallel techniques to compute approximate
betweenness centrality in [16], using a random source selection strategy.
2
While prior approaches approximate centrality scores of all vertices in the
graph, there are no known algorithms to compute the centrality of a single vertex
in time faster than computing the betweenness of all vertices. In this paper, we
present a novel adaptive sampling-based algorithm for approximately computing
betweenness centrality of a given vertex. Our primary result is as follows:
Theorem : For 0 < ² < 0.5, if the centrality of a vertex v is n2 /t for some
constant t ≥ 1, then with probability ≥ 1 − 2² its centrality can be estimated to
within a factor of 1/² with ²t samples of source vertices.
The rest of this paper is organized as follows. We review the currently-known
fastest sequential algorithm by Brandes in Section 2. We present our approximation algorithm based on adaptive sampling and its analysis in Section 3. Section 4
is an experimental study of our approximation technique on several real-world
networks. We conclude with a summary of open problems in Section 5.
2
Exact computation of Betweenness Centrality
Brandes’ algorithm [15] shows how to compute centrality scores of all the vertices
in the graph in the same asymptotic time bounds as n SSSP computations.
2.1
Brandes’ Algorithm
Define the dependency of a source vertex s ∈ V on a vertex v ∈ V as δs∗ (v) =
P
δst (v). Then the betweenness score of v can be then expressed as
t6=s6=v∈V P
BC(v) =
s6=v∈V δs∗ (v). Also, let Ps (v) denote the set of predecessors of a
vertex v on shortest paths from s: Ps (v) = {u ∈ V : hu, vi ∈ E, d(s, v) =
d(s, u) + w(u, v)}. Brandes shows that the dependencies satisfy the following
recursive relation, which is the most crucial step in the algorithm analysis.
Theorem 1. The dependency of s ∈ V on any v ∈ V obeys
δs∗ (v) =
X
w:v∈Ps (w)
λsv
(1 + δs∗ (w))
λsw
First, n SSSP computations are done, one for each s ∈ V . The predecessor
sets Ps (v) are maintained during these computations. Next, for every s ∈ V ,
using the information from the shortest paths tree and predecessor sets along
the paths, compute the dependencies δs∗ (v) for all other v ∈ V . To compute
the centrality value of a vertex v, we finally compute the sum of all dependency
values. The O(n2 ) space requirements can be reduced to O(m+n) by maintaining
a running centrality score. For additional details, we refer the reader to Brandes’
paper [15].
3
3
Adaptive-sampling based approximation
The adaptive sampling technique was introduced by Lipton and Naughton [19]
for estimating the size of the transitive closure of a digraph. Prior to their work,
algorithms for estimating transitive closure were based on randomly sampling
source-vertices, solving the single-source reachability problem for the sampled
vertices, and using this information to estimate the size of the transitive closure.
The Lipton-Naughton algorithm introduces adaptive sampling of source-vertices,
that is, the number of samples varies with the information obtained from each
sample.
In this section, we give an adaptive sampling algorithm for computing betweenness of a given vertex v. It is a sampling algorithm in that it estimates
the centrality by sampling a subset of vertices and performing SSSP computations from these vertices. It is termed adaptive, because the number of samples
required varies with the information obtained from each sample.
The following lemma is easy to see and the proof is omitted.
Lemma 1. BC(v) is zero iff its neighboring vertices induce a clique.
Let
Pai denote the dependency of the vertex vi on v i.e., ai = δvi ∗ (v). Let
A =
ai = BC(v). It is easy to verify that 0 ≤ ai ≤ n − 2 and 0 ≤ A ≤
(n − 1)(n − 2)/2. The quantity we wish to estimate is A. Consider doing so with
the following algorithm:
Algorithm 1 Repeatedly sample a vertex vi ∈ V ; perform SSSP (using BFS or
Dijkstra’s algorithm) from vi and maintain a running sum S of the dependency
scores δvi ∗ (v). Sample until S is greater than cn for some constant c ≥ 2. Let
the total number of samples be k. The estimated betweenness centrality score of
nS
v, BC(v) is given by
.
k
Let Xi be the random variable representing the dependency of a randomly
sampled vertex on v. The probability of an event x is denoted by Pr [ x ]. We
establish the following lemmas to analyze the above algorithm.
Lemma 2. Let E[Xi ] denote the expectation of Xi and V ar[Xi ] denote the variance of Xi . Then, E[Xi ] = A/n, E[Xi 2 ] ≤ A, and V ar[Xi ] ≤ A.
The next lemma is useful in proving a lower bound on the expected number
of samples made before stopping. The proof is presented in the Appendix.
Lemma 3. Let k = ²n2 /A. Then,
Pr [ X1 + X2 + · · · + Xk ≥ cn ] ≤
²
(c − ²)2
Lemma 4. Let k ≥ ²n2 /A and d > 0. Then
à k
"
!
#
n X
1
Pr |
Xi − A| ≥ dA ≤ 2
k i=1
²d
4
Theorem 2. Let à be the estimate of A in the above procedure and let A > 0.
Then for 0 < ² < 0.5 with probability ≥ 1 − 2², Algorithm 1 estimates A to
within a factor of 1/².
Proof. There are two ways that the algorithm can fail: (i) it can stop too early
to guarantee a good error bound, (ii) it can stop after enough samples but with
a bad estimate.
First we claim that the procedure is unlikely to stop with k ≤ n2 /A. We have
that
Pr [ (∃j)(j ≤ k) ∧ (X1 + X2 + · · · + Xj ≥ cn) ] ≤ Pr [ X1 + X2 + · · · + Xk ≥ cn ]
²n2
where k =
, because the event to the right of the inequality implies the
A
event to the left. But by Lemma 3, the right side of this equation is at most
²/(c − ²)2 . Substituting c = 2 and noting that 0 < ² < 0.5, we get that this
probability is less than ².
Next we turn to the accuracy of the estimate. If k = ²n2 /A, by Lemma 4 the
estimate,
k
nX
à =
Xi
k i=1
is within dA of A with probability ≥ 1/(²d2 ). Letting d = 1/², this is just ².
Putting the two ways of failure together, we get that the total probability of
failure is less than ² + (1 − ²)², which is less than 2². Finally, note that if A > 0,
there must be at least one i such that ai > 0, so the algorithm will terminate.
The case when A = 0 (i.e., centrality of v is 0) can be detected using Lemma 1
(before running the algorithm).
An interesting aspect of our theorem is that the sampling is adaptive. usually
such sampling procedures perform a fixed number of samples. Here it is critical
n2
that the algorithm adapts it behavior. Substituting A =
in our analysis we
t
get the following theorem.
Theorem 3. For 0 < ² < 0.5, if the centrality of a vertex v is n2 /t for some
constant t ≥ 1, then with probability ≥ 1 − 2² its centrality can be estimated to
within a factor of 1/² with ²t samples of source vertices.
Although our theoretical result is valid only for high centrality nodes, our
experimental results (presented in the next section) show a similar behavior for
all the vertices.
4
Experimental Study
We assess the quality of the sampling-based approximation algorithm on several
real-world graph instances (see Table 1). We use the parallel centrality analysis
5
toolkit SNAP [20] to compute exact betweenness scores. Since the execution time
and speedup achieved by the approximation approach are directly proportional
to the number of BFS/shortest path computations, we do not report performance
results in this section. For a detailed discussion of exact centrality computation
in parallel, and optimizations for small-world graphs, please refer to [16].
Network data
Label
rand
pref-attach
bio-pin
crawl
cite
road
Network
n
m Details
random graph
2000 7980 synthetic, undirected
preferential attachment
2000 7980 synthetic, undirected
human protein interactions 8503 32,191 undirected
web-crawl (stanford.edu) 9914 36,854 directed
Lederberg citation network 8843 41,601 directed
Rome, Italy road network 3353 4435 weighted, undirected
Table 1. Networks used in the experimental study
Source
[21]
[22]
[23]
[24]
[24, 25]
[26]
We experiment with two synthetic graph instances and four real networks
in this study. rand is an unweighted, undirected random network of 2000 vertices and 7980 edges, generated using the Erdős–Rényi graph model [27]. This
synthetic graph has a low diameter, low clustering, and a Gaussian degree distribution. pref-attach is a synthetic graph generated using the Preferential
attachment model proposed by Barabási and Albert [28]. This model generates
graphs with heavy-tailed degree distributions and scale-free properties. Vertices
are added one at a time, and for each of them, we create a fixed number of
edges connecting to existing vertices, with probability proportional to their degree. bio-pin is a biological network that represents interactions in the human
proteome [29, 23]. This graph is undirected, unweighted and exhibits small-world
characteristics. crawl corresponds to the wb-cs-stanford network in the UF
sparse matrix collection [24]. It is a directed graph, where vertices correspond
to pages in the Stanford Computer Science domain, and edges represent links.
cite is a directed graph from the Pajek network collection [25]. It corresponds
to papers by and citing J. Lederberg (1945-2002). road is a weighted graph of
3353 vertices and 4435 edges that corresponds to a large portion of the road
network of Rome, Italy from 1999 [26]. Vertices correspond to intersections between roads, and edges correspond to roads or road segments. Edge weights are
physical distances in metres. Road networks have more structure and a higher
diameter than the other networks considered in this study.
Methodology
Our goal in this study is to quantify the approximation quality, and so we primarily compare the approximation results to exact scores. We first compute
6
exact centrality scores of all the networks in Table 1. In most data sets, we are
interested in high-centrality vertices, as they are the critical entities and are used
in further analysis. From the exact scores, we identify vertices whose centrality
scores are an order of magnitude greater than the rest of the network. For these
vertices, we study the trade-off between computation and approximation quality by varying the parameter c in Algorithm 1. We also show that it is easy to
estimate scores of low-centrality vertices. We chose small networks for ease of
analysis and visualization, but the approximation algorithm can be effectively
applied to large networks as well (see, for instance, the networks considered in
[16]).
Experiments
Figure 1 plots the distribution of exact and approximate betweenness scores
for the six different test instances. Note that the synthetic networks, rand
and pref-attach show significantly lower variation in exact centrality scores
compared to the real instances. Also, there are a significant percentage of lowcentrality vertices (scores less than, or close to, n) in cite, crawl and bio-pin.
We apply Algorithm 1 to estimate betweenness centrality scores of all the
vertices in the test instances. In order to visualize the data better, we plot a
smoothed curve of the estimated betweenness centrality data that is superimposed with the exact centrality score scatter-plot. We set the parameter c in
n
Algorithm 1 to 5 for these experiments. In addition, we impose a cut-off of 20
on
the number of samples. Observe that in all the networks, the estimated centrality scores are very close to the exact ones, and we are guaranteed to cut down
on the computation by a factor of nearly 20.
To further study the quality of approximation for high-centrality vertices, we
select the top 1% of the vertices (about 30) ordered by exact centrality score in
each network, and compute their estimated centrality scores using the adaptivesampling algorithm. Since the source vertices in the adaptive approach are chosen
randomly, we repeat the experiment five times for each vertex and report the
mean and variance in approximation error. Figure 2 plots the mean percentage
approximation error in the computed scores for these high centrality vertices,
when the value of c (see Algorithm 3) is set to 5. The vertices are sorted by exact
centrality score on the X-axis. The error bars in the charts indicate the variance
in estimated score due to random runs, for each network. For the random graph
instance, the average error is about 5%, while it is roughly around 10% for the
rest of the networks. Except for a few anomalous vertices, the error variance is
within reasonable bounds in all the graph classes.
Figure 3 plots the percentage of BFS/SSSP computations required for approximating the centrality scores, when c is set to 5. This algorithmic count is
an indicator of the amount of work done by the approximation algorithm. The
vertices are ordered again by their exact centrality scores from left to right, with
the vertex with the least score to the left. A common trend we observe across all
graph classes is that the percentage of source vertices decreases as the centrality score increases – this implies that the scores of high centrality vertices can
7
(a) rand
(b) pref-attach
(c) bio-pin
(d) crawl
(e) cite
(f) road
Fig. 1. A scatter plot of exact betweenness scores of all the vertices (in sorted order),
and a line plot of their estimated betweenness scores (the approximate betweenness
scatter data is smoothed by a local smoothing technique using polynomial regression)
8
(a) rand
(b) pref-attach
(c) bio-pin
(d) crawl
(e) cite
(f) road
Fig. 2. Average estimated betweenness error percentage (in comparison to the exact
centrality score) for multiple runs. The adaptive sampling parameter c is set to 5 for
all experiments and the error bars indicate the variance.
9
(a) rand
(b) pref-attach
(c) bio-pin
(d) crawl
(e) cite
(f) road
Fig. 3. The number of samples/SSSP computations as a fraction of n, the total number
of vertices. This algorithmic count is an indicator of the amount of work done by the
approximation algorithm. The adaptive sampling parameter c is set to 5, and the error
bars indicate the variance from 5 runs.
10
be approximated with lesser work using the adaptive sampling approach. Also,
this value is significantly lower for crawl, bio-pin and road compared to other
graph classes.
We can also vary the parameter c, which affects both the percentage of
BFS/SSSP computations and the approximation quality. Table 2 summarizes
the average performance on each graph instance, for different values of c. Taking
only high-centrality vertices into consideration, we report the mean approximation error and the number of samples for each graph instance. As expected, we
find that the error decreases as the parameter c is increased, while the number
of samples increases. Since the highest centrality value is around 10 ∗ n for the
citation network, a significant number of shortest path computations have to
be done even for calculating scores with a reasonable accuracy. But for other
graph instances, particularly the road network, web-crawl and the protein interaction network, c = 5 offers a good trade-off between computation and the
approximation quality.
Network
rand pref-attach bio-pin crawl cite road
t=2
Avg. error 16.28%
29.39%
46.72% 33.69% 32.51% 22.58%
k
Avg.
11.31%
5.36%
1.30% 0.96% 17.00% 0.68 %
n
t=5
Avg. error 6.51%
10.28%
10.49% 10.31% 9.98% 8.79%
k
Avg.
27.37%
12.38%
3.20% 2.42% 43.85% 1.68%
n
t = 10
Avg. error 5.62%
6.13%
7.17% 7.04%
–
7.39%
k
54.51%
24.66%
6.33% 4.89%
–
3.29%
Avg.
n
Table 2. Observed average-case algorithmic counts, as the value of the sampling parameter c is varied. The average error percentage is the deviation of the estimated score
from the exact score, and the nk percentage indicates the number of samples/SSSP computations.
5
Conclusion and Open Problems
We presented a novel approximation algorithm for computing betweenness centrality, of a given vertex, in both weighted and unweighted graphs. Our approximation algorithm is based on an adaptive sampling technique that significantly
reduces the number of single-source shortest path computations for vertices with
high centrality. We conduct an extensive experimental study on real-world graph
instances, and observe that the approximation algorithm performs well on web
crawls, road networks and biological networks.
Approximating the centrality of all vertices in time less than O(nm) for
unweighted graphs and O(nm + n2 log n) for weighted graphs is a challenging
11
open problem. Designing a fully dynamic algorithm for computing betweenness
is very useful.
Acknowledgments
The authors are grateful to the ARC (Algorithms and Randomness Center) of the
College of Computing, Georgia Institute of Technology, for funding this project.
This work was also supported in part by NSF Grants CAREER CCF-0611589,
NSF DBI-0420513, ITR EF/BIO 03-31654, and DARPA Contract NBCH 30390004.
Kamesh Madduri’s work is supported in part by the NASA Graduate Student
Researcher Program Fellowship (NASA NP-2005-07-375-HQ). We thank Richard
Lipton for helpful discussions.
Appendix
Lemma 3
Let k = ²n2 /A. Then,
Pr [ X1 + X2 + · · · + Xk ≥ cn ] ≤
²
(c − ²)2
Proof. We have
·µ
¶
µ
A
Pr [ X1 + · · · + Xk ≥ cn ] = Pr
X1 −
+ · · · + Xk −
n
¶
µ
·µ
A
+ · · · + Xk −
= Pr
X1 −
n
·
¸
X
A
≤
Pr Xi −
≥ (c − ²)n
n
i
X
1
≤
V ar[Xi ]
(c − ²)2 n2
i
X
1
=
V ar[Xi ]
2
2
(c − ²) n i
¶
¸
A
kA
≥ cn −
n
n
¶
¸
A
≥ cn − ²n
n
1
kA
(c − ²)2 n2
²
=
(c − ²)2
≤
Note that we have used Chebychev’s inequality and union bounds in the
above proof. We bound the error in the estimated value of A with the following
lemma.
12
Lemma 4
Let k ≥ ²n2 /A and d > 0. Then
!
#
"
à k
1
n X
Xi − A| ≥ dA ≤ 2
Pr |
k i=1
²d
Proof.
"
Pr
n
|
k
à k
X
i=1
!
Xi
#
− A| ≥ t
#
" Ã k
!
X
kt
k
= Pr |
Xi − A| ≥
n
n
i=1
" Ã k
!
#
X
1
kt
= Pr |
Xi − A | ≥
n
n
i=1
≤
Let k = λ
to
which is just
n2
k·V ar[Xi ]
k 2 t2
n2
, where λ ≥ ². Then the above probability is less than or equal
A
n2
n2
k·V ar[Xi ] ≤ n2 A
2
2
k t
λ A t2
A2
. Setting Ad = t gives
λt2
1
1
≤ 2
2
λd
²d
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